:: HILBASIS semantic presentation

REAL is V72() V73() V74() V78() V292() V293() V295() set
NAT is non empty non trivial ordinal non finite cardinal limit_cardinal V72() V73() V74() V75() V76() V77() V78() V290() V292() Element of bool REAL
bool REAL is non empty set
NAT is non empty non trivial ordinal non finite cardinal limit_cardinal V72() V73() V74() V75() V76() V77() V78() V290() V292() set
bool NAT is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
COMPLEX is V72() V78() set
RAT is V72() V73() V74() V75() V78() set
INT is V72() V73() V74() V75() V76() V78() set
[:REAL,REAL:] is Relation-like V62() V63() V64() set
bool [:REAL,REAL:] is non empty set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
{{},1} is non empty finite V34() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
K377() is set
bool K377() is non empty set
K378() is Element of bool K377()
2 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:NAT,NAT:] is non empty non trivial non finite set
[:NAT,REAL:] is Relation-like V62() V63() V64() set
bool [:NAT,REAL:] is non empty set
K239(1,NAT) is M10( NAT )
[:COMPLEX,COMPLEX:] is Relation-like V62() set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like V62() set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:[:REAL,REAL:],REAL:] is Relation-like V62() V63() V64() set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is Relation-like RAT -valued V62() V63() V64() set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued V62() V63() V64() set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is Relation-like RAT -valued INT -valued V62() V63() V64() set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued V62() V63() V64() set
bool [:[:INT,INT:],INT:] is non empty set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
K859() is set
K637() is Relation-like NAT -defined Function-like total set
3 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Seg 1 is non empty trivial finite 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
{1} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
Seg 2 is non empty finite 2 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
{1,2} is non empty finite V34() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V49() V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of NAT
Bags {} is functional non empty Element of bool (Bags {})
Bags {} is non empty set
bool (Bags {}) is non empty set
{{}} is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
R is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
rng R is finite set
X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
rng X is finite set
(rng R) \/ (rng X) is finite set
R ^ X is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
f0 is Relation-like Function-like set
dom f0 is set
R * f0 is Relation-like NAT -defined Function-like finite finite-support set
X * f0 is Relation-like NAT -defined Function-like finite finite-support set
(R ^ X) * f0 is Relation-like NAT -defined Function-like finite finite-support set
rng (R ^ X) is finite set
dom (X * f0) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom X is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
cR is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len cR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
S ^ cR is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom ((R ^ X) * f0) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (R ^ X) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (R * f0) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom R is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
tcR is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len tcR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom tcR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(S ^ cR) . tcPR is set
S . tcPR is set
R . tcPR is set
f0 . (R . tcPR) is set
(R ^ X) . tcPR is set
f0 . ((R ^ X) . tcPR) is set
tcR . tcPR is set
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
(len R) + 0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR . tcPR is set
(R ^ X) . tcPR is set
f0 . ((R ^ X) . tcPR) is set
X . 0X is set
f0 . (X . 0X) is set
cR . 0X is set
(S ^ cR) . tcPR is set
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
len (R ^ X) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len S) + (len cR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (S ^ cR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
<*{},{}*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
<*<*{},{}*>*> is Relation-like NAT -defined Function-like constant non empty trivial Function-yielding V22() finite 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support set
R is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
decomp R is Relation-like NAT -defined K239(2,(Bags 0)) -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of K239(2,(Bags 0))
Bags 0 is functional non empty Element of bool (Bags 0)
Bags 0 is non empty set
bool (Bags 0) is non empty set
K239(2,(Bags 0)) is M10( Bags 0)
EmptyBag {} is Relation-like non-empty empty-yielding {} -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of Bags {}
{} --> 0 is Relation-like non-empty empty-yielding {} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one constant functional empty total quasi_total Function-yielding V22() ordinal T-Sequence-like natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of bool [:{},NAT:]
[:{},NAT:] is Relation-like RAT -valued INT -valued V62() V63() V64() V65() set
bool [:{},NAT:] is non empty set
{0} is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
[:{},{0}:] is Relation-like RAT -valued INT -valued finite V62() V63() V64() V65() set
divisors R is Relation-like NAT -defined Bags 0 -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support FinSequence of Bags 0
<*{}*> is Relation-like NAT -defined Function-like constant non empty trivial Function-yielding V22() finite 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support set
len (divisors R) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
dom (divisors R) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
dom (decomp R) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
(decomp R) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
(decomp R) /. 1 is Relation-like NAT -defined Bags 0 -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags 0))
len (decomp R) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
f0 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
f0 | R is Relation-like R -defined X -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
tcR is set
{ b₁ where b₁ is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT : not R <= b₁ } is set
cR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
{ b₁ where b₁ is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT : not X <= b₁ } is set
X is set
R is set
f0 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
f0 | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
tcR is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
tcR | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
cR is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
S is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
dom cR is Element of bool R
bool R is non empty set
dom S is Element of bool R
tcPR is set
cR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR is set
cR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR is set
X is set
R is set
f0 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
f0 | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
tcR is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
tcR | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
f0 -' tcR is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(f0 -' tcR) | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
f0 + tcR is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(f0 + tcR) | R is Relation-like R -defined X -defined RAT -valued Function-like V62() V63() V64() V65() set
cR is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
S is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cR -' S is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cR + S is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
dom f0 is Element of bool X
bool X is non empty set
dom (f0 | R) is Element of bool R
bool R is non empty set
X /\ R is set
dom tcR is Element of bool X
dom (tcR | R) is Element of bool R
dom (f0 + tcR) is Element of bool X
dom ((f0 + tcR) | R) is Element of bool R
tcPR is set
((f0 + tcR) | R) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(f0 + tcR) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(f0 . tcPR) + (tcR . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR . tcPR) + (tcR . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR . tcPR) + (S . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR + S) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (f0 -' tcR) is Element of bool X
dom ((f0 -' tcR) | R) is Element of bool R
tcPR is set
((f0 -' tcR) | R) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(f0 -' tcR) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(f0 . tcPR) -' (tcR . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR . tcPR) -' (tcR . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR . tcPR) -' (S . tcPR) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(cR -' S) . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (cR -' S) is Element of bool R
dom (cR + S) is Element of bool R
R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
R + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (R + 1) is non empty set
Bags (R + 1) is functional non empty Element of bool (Bags (R + 1))
bool (Bags (R + 1)) is non empty set
f0 is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
R .--> X is Relation-like {R} -defined RAT -valued INT -valued Function-like one-to-one finite V62() V63() V64() V65() finite-support set
{R} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
{R} --> X is Relation-like {R} -defined RAT -valued INT -valued {X} -valued Function-like constant non empty total quasi_total finite V62() V63() V64() V65() finite-support Element of bool [:{R},{X}:]
{X} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
[:{R},{X}:] is Relation-like RAT -valued INT -valued non empty finite V62() V63() V64() V65() set
bool [:{R},{X}:] is non empty finite V34() set
f0 +* (R .--> X) is Relation-like RAT -valued Function-like V62() V63() V64() V65() set
dom f0 is finite Element of bool R
bool R is non empty finite V34() set
dom (f0 +* (R .--> X)) is set
dom (R .--> X) is trivial finite V34() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool {R}
bool {R} is non empty finite V34() set
(dom f0) \/ (dom (R .--> X)) is finite set
(dom f0) \/ {R} is non empty finite set
R \/ {R} is non empty finite set
succ R is set
cR is Relation-like R + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (R + 1)
cR | R is Relation-like R -defined R + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cR . R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (cR | R) is finite Element of bool R
(R + 1) /\ R is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() set
(succ R) /\ R is finite set
(R \/ {R}) /\ R is finite set
S is set
(cR | R) . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R .--> X) . R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR is Relation-like R + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (R + 1)
tcR | R is Relation-like R -defined R + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
tcR . R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR is Relation-like R + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (R + 1)
cR | R is Relation-like R -defined R + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cR . R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom tcR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (R + 1)
bool (R + 1) is non empty finite V34() set
S is set
succ R is set
{R} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
R \/ {R} is non empty finite set
tcR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R + 1) /\ R is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() set
dom f0 is finite Element of bool R
bool R is non empty finite V34() set
tcR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom cR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (R + 1)
R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
EmptyBag (R + 1) is Relation-like R + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (R + 1)
Bags (R + 1) is non empty set
Bags (R + 1) is functional non empty Element of bool (Bags (R + 1))
bool (Bags (R + 1)) is non empty set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
(R,0,(EmptyBag R)) is Relation-like R + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (R + 1)
R + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (R + 1) is non empty set
Bags (R + 1) is functional non empty Element of bool (Bags (R + 1))
bool (Bags (R + 1)) is non empty set
X is set
succ R is set
{R} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
R \/ {R} is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
(EmptyBag (R + 1)) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(EmptyBag R) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,0,(EmptyBag R)) | R is Relation-like R -defined R + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
((R,0,(EmptyBag R)) | R) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,0,(EmptyBag R)) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(EmptyBag (R + 1)) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,0,(EmptyBag R)) . R is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,0,(EmptyBag R)) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (EmptyBag (R + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (R + 1)
bool (R + 1) is non empty finite V34() set
dom (R,0,(EmptyBag R)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (R + 1)
bool (R + 1) is non empty finite V34() set
R is ordinal set
Bags R is functional non empty Element of bool (Bags R)
Bags R is non empty set
bool (Bags R) is non empty set
f0 is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
X is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
divisors X is Relation-like NAT -defined Bags R -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support FinSequence of Bags R
rng (divisors X) is functional non empty finite Element of bool (Bags R)
bool (Bags R) is non empty set
BagOrder R is Relation-like Bags R -defined Bags R -valued total quasi_total V53() V56() V60() being_linear-order Element of bool [:(Bags R),(Bags R):]
[:(Bags R),(Bags R):] is Relation-like non empty set
bool [:(Bags R),(Bags R):] is non empty set
tcR is functional non empty finite Element of bool (Bags R)
SgmX ((BagOrder R),tcR) is Relation-like NAT -defined Bags R -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support FinSequence of Bags R
field (BagOrder R) is set
R is set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
X is Element of R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
R is non empty set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
support (R,X) is finite Element of bool R
bool R is non empty set
{X} is non empty trivial finite 1 -element Element of bool R
f0 is set
((EmptyBag R) +* (X,1)) . f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(EmptyBag R) . f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R --> 0 is Relation-like R -defined NAT -valued RAT -valued INT -valued Function-like constant non empty total quasi_total Function-yielding V22() V62() V63() V64() V65() Element of bool [:R,NAT:]
[:R,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:R,NAT:] is non empty non trivial non finite set
{0} is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
[:R,{0}:] is Relation-like RAT -valued INT -valued non empty V62() V63() V64() V65() set
dom (EmptyBag R) is Element of bool R
(R,X) . f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is non empty set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(R,X) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R --> 0 is Relation-like R -defined NAT -valued RAT -valued INT -valued Function-like constant non empty total quasi_total Function-yielding V22() V62() V63() V64() V65() Element of bool [:R,NAT:]
[:R,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:R,NAT:] is non empty non trivial non finite set
{0} is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
[:R,{0}:] is Relation-like RAT -valued INT -valued non empty V62() V63() V64() V65() set
dom (R --> 0) is non empty Element of bool R
bool R is non empty set
(R --> 0) +* (X,1) is Relation-like R -defined NAT -valued RAT -valued Function-like non empty total quasi_total V62() V63() V64() V65() Element of bool [:R,NAT:]
((R --> 0) +* (X,1)) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 is Element of R
(R,X) . f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(EmptyBag R) . f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is non empty set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
f0 is Element of R
(R,f0) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (f0,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(R,f0) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is non empty ordinal set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
Bags R is non empty set
Bags R is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
f0 is non empty non trivial unital right_unital well-unital left_unital doubleLoopStr
the carrier of f0 is non empty non trivial set
[:R, the carrier of f0:] is Relation-like non empty set
bool [:R, the carrier of f0:] is non empty set
tcR is Relation-like R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:R, the carrier of f0:]
eval ((R,X),tcR) is Element of the carrier of f0
tcR . X is Element of the carrier of f0
support (R,X) is finite Element of bool R
bool R is non empty set
{X} is non empty trivial finite 1 -element Element of bool R
power f0 is Relation-like [: the carrier of f0,NAT:] -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of f0,NAT:], the carrier of f0:]
[: the carrier of f0,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
[:[: the carrier of f0,NAT:], the carrier of f0:] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of f0,NAT:], the carrier of f0:] is non empty non trivial non finite set
(R,X) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(power f0) . ((tcR . X),((R,X) . X)) is Element of the carrier of f0
0 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(power f0) . ((tcR . X),(0 + 1)) is Element of the carrier of f0
cR is Element of the carrier of f0
(power f0) . (cR,0) is Element of the carrier of f0
((power f0) . (cR,0)) * cR is Element of the carrier of f0
1_ f0 is Element of the carrier of f0
K409(f0) is Element of the carrier of f0
the OneF of f0 is Element of the carrier of f0
(1_ f0) * cR is Element of the carrier of f0
R is set
Bags R is functional non empty Element of bool (Bags R)
Bags R is non empty set
bool (Bags R) is non empty set
f0 is non empty unital multLoopStr_0
the carrier of f0 is non empty set
0_ (R,f0) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
[:(Bags R), the carrier of f0:] is Relation-like non empty set
bool [:(Bags R), the carrier of f0:] is non empty set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ f0 is Element of the carrier of f0
(0_ (R,f0)) +* ((R,X),(1_ f0)) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
R is set
X is non empty non trivial unital doubleLoopStr
1_ X is Element of the carrier of X
the carrier of X is non empty non trivial set
0. X is V102(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
f0 is Element of R
(R,f0,X) is Relation-like Bags R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of X:]
Bags R is functional non empty Element of bool (Bags R)
Bags R is non empty set
bool (Bags R) is non empty set
[:(Bags R), the carrier of X:] is Relation-like non empty set
bool [:(Bags R), the carrier of X:] is non empty set
0_ (R,X) is Relation-like Bags R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of X:]
(R,f0) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (f0,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (R,X)) +* ((R,f0),(1_ X)) is Relation-like Bags R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of X:]
(R,f0,X) . (R,f0) is Element of the carrier of X
dom (0_ (R,X)) is functional non empty Element of bool (Bags R)
bool (Bags R) is non empty set
(Bags R) --> (0. X) is Relation-like Bags R -defined the carrier of X -valued Function-like constant non empty total quasi_total Element of bool [:(Bags R), the carrier of X:]
{(0. X)} is non empty trivial finite 1 -element set
[:(Bags R),{(0. X)}:] is Relation-like non empty set
dom ((Bags R) --> (0. X)) is functional non empty Element of bool (Bags R)
tcR is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(R,f0,X) . tcR is Element of the carrier of X
(0_ (R,X)) . tcR is Element of the carrier of X
R is set
Bags R is functional non empty Element of bool (Bags R)
Bags R is non empty set
bool (Bags R) is non empty set
X is Element of R
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
{(R,X)} is functional non empty trivial finite 1 -element Element of bool (Bags R)
bool (Bags R) is non empty set
f0 is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive right_unital well-unital left_unital V275() V276() V277() V278() doubleLoopStr
(R,X,f0) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
the carrier of f0 is non empty non trivial set
[:(Bags R), the carrier of f0:] is Relation-like non empty set
bool [:(Bags R), the carrier of f0:] is non empty set
0_ (R,f0) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
1_ f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
K409(f0) is V102(f0) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
the OneF of f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
(0_ (R,f0)) +* ((R,X),(1_ f0)) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
Support (R,X,f0) is functional Element of bool (Bags R)
tcR is set
cR is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(R,X,f0) . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
(0_ (R,f0)) . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
0. f0 is V102(f0) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
the ZeroF of f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
(R,X,f0) . tcR is set
R is ordinal set
Bags R is functional non empty Element of bool (Bags R)
Bags R is non empty set
bool (Bags R) is non empty set
f0 is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive right_unital well-unital left_unital V275() V276() V277() V278() doubleLoopStr
X is Element of R
(R,X,f0) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
the carrier of f0 is non empty non trivial set
[:(Bags R), the carrier of f0:] is Relation-like non empty set
bool [:(Bags R), the carrier of f0:] is non empty set
0_ (R,f0) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags R), the carrier of f0:]
(R,X) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
EmptyBag R is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags R
(EmptyBag R) +* (X,1) is Relation-like R -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
K409(f0) is V102(f0) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
the OneF of f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of f0
(0_ (R,f0)) +* ((R,X),(1_ f0)) is Relation-like Bags R -defined the carrier of f0 -valued Function-like non empty total quasi_total Element of bool [:(Bags R), the carrier of f0:]
Support (R,X,f0) is functional Element of bool (Bags R)
bool (Bags R) is non empty set
{(R,X)} is functional non empty trivial finite 1 -element Element of bool (Bags R)
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive right_unital well-unital left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty non trivial set
X is non empty set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
f0 is Element of X
(X,f0,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
(X,f0) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (f0,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0_ (X,R)) +* ((X,f0),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
tcR is Element of X
(X,tcR,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
(X,tcR) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (tcR,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,tcR),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
(X,tcR,R) . (X,f0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive left-distributive distributive V275() V276() V277() V278() doubleLoopStr
Polynom-Ring R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring R) is non empty set
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
- X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
- f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
cR is set
f0 - f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
f0 + (- f0) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
(f0 - f0) . cR is set
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(f0 . S) - (f0 . S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0_. R is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
NAT --> (0. R) is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:NAT, the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:NAT,{(0. R)}:] is Relation-like non empty non trivial non finite set
(0_. R) . cR is set
tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
- tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
dom (0_. R) is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
dom (f0 - f0) is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
X + cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
tcR + (- tcR) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
0. (Polynom-Ring R) is V102( Polynom-Ring R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
the ZeroF of (Polynom-Ring R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
- cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive left-distributive distributive V275() V276() V277() V278() doubleLoopStr
Polynom-Ring R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring R) is non empty set
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
X - f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
tcR - cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
- cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
tcR + (- cR) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
- f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
X + (- f0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring R) is non empty set
bool the carrier of (Polynom-Ring R) is non empty set
X is non empty Element of bool the carrier of (Polynom-Ring R)
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
{ b₁ where b₁ is Element of X : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in X or len b₂ <= len b₃ ) } is set
bool X is non empty set
cR is Element of X
S is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
[:X,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:X,NAT:] is non empty non trivial non finite set
cR is Relation-like X -defined NAT -valued Function-like non empty total quasi_total V62() V63() V64() V65() Element of bool [:X,NAT:]
f0 is non empty Element of bool X
cR .: f0 is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
min (cR .: f0) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() set
cR .: X is V72() V73() V74() V75() V76() V77() V292() Element of bool NAT
dom cR is non empty Element of bool X
S is set
cR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
tcPR is Element of X
S is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR . S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len 0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X is set
S is Element of X
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring R) is non empty set
bool the carrier of (Polynom-Ring R) is non empty set
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is non empty Element of bool the carrier of (Polynom-Ring R)
(R,X) is non empty Element of bool X
bool X is non empty set
{ b₁ where b₁ is Element of X : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in X or len b₂ <= len b₃ ) } is set
f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len tcR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR is Element of X
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0_. R is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
NAT --> (0. R) is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:NAT, the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:NAT,{(0. R)}:] is Relation-like non empty non trivial non finite set
(0_. R) +* (X,f0) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
cR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
tcR . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(NAT --> (0. R)) . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom (0_. R) is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
(NAT --> (0. R)) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
S is set
tcR . S is set
cR . S is set
tcR . S is set
cR . S is set
dom tcR is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
dom cR is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,f0) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
len (R,X,f0) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
(R,X,f0) . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,f0) . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
(R,X,f0) . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,f0) . X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 + X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
(R,X,tcR) *' cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
((R,X,tcR) *' cR) . (f0 + X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcR * (cR . f0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(f0 + X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
1 + f0 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
X + (1 + f0) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
tcPR is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len 0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR ^ 0X is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S ^ C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom 0X is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom tcPR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN - 1 is V42() V43() V44() ext-real set
CN -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . CN is set
S . CN is set
(R,X,tcR) . (CN -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((f0 + X) + 1) -' CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . (((f0 + X) + 1) -' CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,X,tcR) . (CN -' 1)) * (cR . (((f0 + X) + 1) -' CN)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) * (cR . (((f0 + X) + 1) -' CN)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
1 + CN is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
X + (1 + CN) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(X + (1 + CN)) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
X + CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X + CN) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((X + CN) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom C is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
C . CN is set
0X . (1 + CN) is set
S . (X + (1 + CN)) is set
(R,X,tcR) . ((X + (1 + CN)) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((f0 + X) + 1) -' (X + (1 + CN)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . (((f0 + X) + 1) -' (X + (1 + CN))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,X,tcR) . ((X + (1 + CN)) -' 1)) * (cR . (((f0 + X) + 1) -' (X + (1 + CN)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) * (cR . (((f0 + X) + 1) -' (X + (1 + CN)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S . 1 is set
0X . 1 is set
S . (X + 1) is set
(X + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,X,tcR) . ((X + 1) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((f0 + X) + 1) -' (X + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . (((f0 + X) + 1) -' (X + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,X,tcR) . ((X + 1) -' 1)) * (cR . (((f0 + X) + 1) -' (X + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) . X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 + (X + 1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(f0 + (X + 1)) -' (X + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . ((f0 + (X + 1)) -' (X + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,X,tcR) . X) * (cR . ((f0 + (X + 1)) -' (X + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,X,tcR) . X) * (cR . f0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*(tcR * (cR . f0))*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(tcR * (cR . f0)) + (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) + (tcR * (cR . f0)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 + X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
cR *' (R,X,tcR) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
(cR *' (R,X,tcR)) . (f0 + X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . f0) * tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(f0 + X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
f0 + (X + 1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
tcPR is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len 0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR ^ 0X is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S ^ C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom 0X is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
X - 1 is V42() V43() V44() ext-real set
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom C is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
1 + CN is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
f0 + CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(f0 + CN) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
f0 + (1 + CN) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((f0 + X) + 1) -' (f0 + (1 + CN)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((f0 + X) + 1) - (f0 + (1 + CN)) is V42() V43() V44() ext-real set
X - CN is V42() V43() V44() ext-real set
C . CN is set
0X . (1 + CN) is set
S . (f0 + (1 + CN)) is set
(f0 + (1 + CN)) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . ((f0 + (1 + CN)) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) . (((f0 + X) + 1) -' (f0 + (1 + CN))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . ((f0 + (1 + CN)) -' 1)) * ((R,X,tcR) . (((f0 + X) + 1) -' (f0 + (1 + CN)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . ((f0 + (1 + CN)) -' 1)) * (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom tcPR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN + X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((f0 + X) + 1) - CN is V42() V43() V44() ext-real set
((f0 + X) + 1) -' CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . CN is set
S . CN is set
CN -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . (CN -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) . (((f0 + X) + 1) -' CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . (CN -' 1)) * ((R,X,tcR) . (((f0 + X) + 1) -' CN)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . (CN -' 1)) * (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S . 1 is set
0X . 1 is set
f0 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (f0 + 1) is set
(f0 + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . ((f0 + 1) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((f0 + X) + 1) -' (f0 + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,X,tcR) . (((f0 + X) + 1) -' (f0 + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . ((f0 + 1) -' 1)) * ((R,X,tcR) . (((f0 + X) + 1) -' (f0 + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
X + (f0 + 1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(X + (f0 + 1)) -' (f0 + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,X,tcR) . ((X + (f0 + 1)) -' (f0 + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . f0) * ((R,X,tcR) . ((X + (f0 + 1)) -' (f0 + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(R,X,tcR) . X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cR . f0) * ((R,X,tcR) . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*((cR . f0) * tcR)*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((cR . f0) * tcR) + (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) + ((cR . f0) * tcR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
X is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
X *' f0 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len (X *' f0) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len f0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len X) + (len f0) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len X) + (len f0)) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len X) + (len f0)) - 1 is V42() V43() V44() ext-real set
tcR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
tcR + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(X *' f0) . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len cR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom cR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(len f0) - 1 is V42() V43() V44() ext-real set
(len X) + ((len f0) - 1) is V42() V43() V44() ext-real set
tcR - ((len f0) - 1) is V42() V43() V44() ext-real set
tcR - (len f0) is V42() V43() V44() ext-real set
(tcR - (len f0)) + 1 is V42() V43() V44() ext-real set
- ((tcR - (len f0)) + 1) is V42() V43() V44() ext-real set
- (len X) is V42() V43() V44() ext-real non positive set
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR . S is set
S -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
X . (S -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(tcR + 1) -' S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
f0 . ((tcR + 1) -' S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X . (S -' 1)) * (f0 . ((tcR + 1) -' S)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Seg (len cR) is finite len cR -element V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(tcR + 1) - S is V42() V43() V44() ext-real set
S - 1 is V42() V43() V44() ext-real set
- (S - 1) is V42() V43() V44() ext-real set
1 - S is V42() V43() V44() ext-real set
((len f0) - 1) - tcR is V42() V43() V44() ext-real set
tcR + (1 - S) is V42() V43() V44() ext-real set
((len f0) - 1) + 1 is V42() V43() V44() ext-real set
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
X is non empty doubleLoopStr
the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
bool the carrier of X is non empty set
f0 is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
tcR is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
tcR .: f0 is non empty Element of bool the carrier of X
tcPR is Element of the carrier of X
0X is set
tcR . 0X is set
rng tcR is non empty Element of bool the carrier of X
dom tcR is non empty Element of bool the carrier of R
S is Element of the carrier of X
C is set
tcR . C is set
CN is Element of the carrier of R
S is Element of the carrier of R
CN * S is Element of the carrier of R
S * tcPR is Element of the carrier of X
tcR . (CN * S) is Element of the carrier of X
tcPR is Element of the carrier of X
0X is set
tcR . 0X is set
S is Element of the carrier of X
C is set
tcR . C is set
S is Element of the carrier of R
CN is Element of the carrier of R
S * CN is Element of the carrier of R
tcPR * S is Element of the carrier of X
tcR . (S * CN) is Element of the carrier of X
S is Element of the carrier of X
tcPR is Element of the carrier of X
0X is set
tcR . 0X is set
S is set
tcR . S is set
C is Element of the carrier of R
CN is Element of the carrier of R
C + CN is Element of the carrier of R
S + tcPR is Element of the carrier of X
tcR . (C + CN) is Element of the carrier of X
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
X is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. X is V102(X) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
the ZeroF of X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
f0 is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
f0 . (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(0. R) + (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . ((0. R) + (0. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(f0 . (0. R)) + (f0 . (0. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
- (f0 . (0. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
((f0 . (0. R)) + (f0 . (0. R))) + (- (f0 . (0. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(f0 . (0. R)) + (- (f0 . (0. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(f0 . (0. R)) + ((f0 . (0. R)) + (- (f0 . (0. R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(f0 . (0. R)) + (0. X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
X is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of X is non empty set
bool the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
[: the carrier of R, the carrier of R, the carrier of R:] is non empty set
f0 is non empty Element of bool the carrier of R
tcR is non empty Element of bool the carrier of X
cR is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . (Sum S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
tcPR is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR
len tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom tcPR is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
Sum tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
0X is Relation-like NAT -defined [: the carrier of R, the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of [: the carrier of R, the carrier of R, the carrier of R:]
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Seg S is finite S -element V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
S | (Seg S) is Relation-like NAT -defined Seg S -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
tcPR | (Seg S) is Relation-like NAT -defined Seg S -defined NAT -defined the carrier of X -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of X:]
[:NAT, the carrier of X:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of X:] is non empty non trivial non finite set
S + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Seg (S + 1) is non empty finite S + 1 -element S + 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
S + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S | (Seg (S + 1)) is Relation-like NAT -defined Seg (S + 1) -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of R:]
tcPR | (Seg (S + 1)) is Relation-like NAT -defined Seg (S + 1) -defined NAT -defined the carrier of X -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of X:]
C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . (Sum C) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
CN is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
S . (S + 1) is set
S /. (S + 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcPR . (S + 1) is set
tcPR /. (S + 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cn is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
<*(S . (S + 1))*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
cn ^ <*(S . (S + 1))*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
Sum cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Sum cn) + (S /. (S + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . ((Sum cn) + (S /. (S + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . (Sum cn) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . (S /. (S + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(cR . (Sum cn)) + (cR . (S /. (S + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cn is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR
Sum cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(Sum cn) + (cR . (S /. (S + 1))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
0X /. (S + 1) is Element of [: the carrier of R, the carrier of R, the carrier of R:]
(0X /. (S + 1)) `1_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0X /. (S + 1)) `2_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0X /. (S + 1)) `3_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3)) * ((0X /. (S + 1)) `3_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . ((((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3)) * ((0X /. (S + 1)) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(Sum cn) + (cR . ((((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3)) * ((0X /. (S + 1)) `3_3))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . (((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . ((0X /. (S + 1)) `3_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(cR . (((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3))) * (cR . ((0X /. (S + 1)) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(Sum cn) + ((cR . (((0X /. (S + 1)) `1_3) * ((0X /. (S + 1)) `2_3))) * (cR . ((0X /. (S + 1)) `3_3))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . ((0X /. (S + 1)) `1_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
cR . ((0X /. (S + 1)) `2_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(cR . ((0X /. (S + 1)) `1_3)) * (cR . ((0X /. (S + 1)) `2_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
((cR . ((0X /. (S + 1)) `1_3)) * (cR . ((0X /. (S + 1)) `2_3))) * (cR . ((0X /. (S + 1)) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(Sum cn) + (((cR . ((0X /. (S + 1)) `1_3)) * (cR . ((0X /. (S + 1)) `2_3))) * (cR . ((0X /. (S + 1)) `3_3))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
(Sum cn) + (tcPR /. (S + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
<*(tcPR /. (S + 1))*> is Relation-like NAT -defined the carrier of X -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of X *
the carrier of X * is functional non empty FinSequence-membered M10( the carrier of X)
cn ^ <*(tcPR /. (S + 1))*> is Relation-like NAT -defined the carrier of X -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of X
Sum (cn ^ <*(tcPR /. (S + 1))*>) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
Seg (len S) is finite len S -element V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
S | (Seg (len S)) is Relation-like NAT -defined Seg (len S) -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
Seg (len tcPR) is finite len tcPR -element V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
tcPR | (Seg (len tcPR)) is Relation-like NAT -defined Seg (len tcPR) -defined NAT -defined the carrier of X -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of X:]
[:NAT, the carrier of X:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of X:] is non empty non trivial non finite set
Seg 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty proper Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal 0 -element {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of bool NAT
S | (Seg 0) is Relation-like non-empty empty-yielding NAT -defined Seg 0 -defined NAT -defined RAT -valued the carrier of R -valued Function-like one-to-one constant functional empty proper Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of bool [:NAT, the carrier of R:]
tcPR | (Seg 0) is Relation-like non-empty empty-yielding NAT -defined Seg 0 -defined NAT -defined RAT -valued the carrier of X -valued Function-like one-to-one constant functional empty proper Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of bool [:NAT, the carrier of X:]
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . (Sum S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
C is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
<*> the carrier of R is Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
Sum (<*> the carrier of R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . (Sum (<*> the carrier of R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR . (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
0. X is V102(X) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
the ZeroF of X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
<*> the carrier of X is Relation-like non-empty empty-yielding NAT -defined the carrier of X -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of X *
the carrier of X * is functional non empty FinSequence-membered M10( the carrier of X)
Sum (<*> the carrier of X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
R is non empty doubleLoopStr
the carrier of R is non empty set
X is non empty doubleLoopStr
the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
f0 is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
f0 " is Relation-like the carrier of X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of X, the carrier of R:]
[: the carrier of X, the carrier of R:] is Relation-like non empty set
bool [: the carrier of X, the carrier of R:] is non empty set
rng f0 is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
1_ X is Element of the carrier of X
(f0 ") . (1_ X) is Element of the carrier of R
1_ R is Element of the carrier of R
f0 . (1_ R) is Element of the carrier of X
(f0 ") . (f0 . (1_ R)) is Element of the carrier of R
f0 " is Relation-like Function-like set
(f0 ") . (f0 . (1_ R)) is set
tcR is Element of the carrier of X
cR is Element of the carrier of X
tcR + cR is Element of the carrier of X
(f0 ") . (tcR + cR) is Element of the carrier of R
(f0 ") . tcR is Element of the carrier of R
(f0 ") . cR is Element of the carrier of R
((f0 ") . tcR) + ((f0 ") . cR) is Element of the carrier of R
tcR * cR is Element of the carrier of X
(f0 ") . (tcR * cR) is Element of the carrier of R
((f0 ") . tcR) * ((f0 ") . cR) is Element of the carrier of R
S is set
f0 . S is set
tcPR is Element of the carrier of R
f0 . tcPR is Element of the carrier of X
(f0 ") . (f0 . tcPR) is set
0X is set
f0 . 0X is set
S is Element of the carrier of R
f0 . S is Element of the carrier of X
(f0 ") . (f0 . S) is set
tcPR * S is Element of the carrier of R
f0 . (tcPR * S) is Element of the carrier of X
(f0 ") . (f0 . (tcPR * S)) is Element of the carrier of R
(f0 ") . (f0 . (tcPR * S)) is set
tcPR + S is Element of the carrier of R
f0 . (tcPR + S) is Element of the carrier of X
(f0 ") . (f0 . (tcPR + S)) is Element of the carrier of R
(f0 ") . (f0 . (tcPR + S)) is set
[#] X is non empty non proper Element of bool the carrier of X
rng (f0 ") is non empty Element of bool the carrier of R
bool the carrier of R is non empty set
[#] R is non empty non proper Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
X is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
f0 is non empty Element of bool the carrier of R
f0 -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
tcR is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
tcR .: (f0 -Ideal) is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
tcR .: f0 is non empty Element of bool the carrier of X
(tcR .: f0) -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of X
dom tcR is non empty Element of bool the carrier of R
cR is set
S is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR .: f0
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
[: the carrier of X, the carrier of X, the carrier of X:] is non empty set
tcPR is Relation-like NAT -defined [: the carrier of X, the carrier of X, the carrier of X:] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of [: the carrier of X, the carrier of X, the carrier of X:]
rng tcR is non empty Element of bool the carrier of X
tcR " is Relation-like Function-like set
tcR " is Relation-like the carrier of X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of X, the carrier of R:]
[: the carrier of X, the carrier of R:] is Relation-like non empty set
bool [: the carrier of X, the carrier of R:] is non empty set
(tcR ") .: (tcR .: f0) is set
tcR " (tcR .: f0) is set
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
len 0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom 0X is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(tcR ") . cR is set
Sum 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcR . ((tcR ") . cR) is set
cR is set
S is set
tcR . S is set
tcPR is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of f0
Sum tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[: the carrier of R, the carrier of R, the carrier of R:] is non empty set
0X is Relation-like NAT -defined [: the carrier of R, the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of [: the carrier of R, the carrier of R, the carrier of R:]
len tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S is Relation-like NAT -defined the carrier of X -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of tcR .: f0
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom S is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
Sum S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of X
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
X is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of X is non empty set
[: the carrier of R, the carrier of X:] is Relation-like non empty set
bool [: the carrier of R, the carrier of X:] is non empty set
f0 is Relation-like the carrier of R -defined the carrier of X -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of X:]
rng f0 is non empty Element of bool the carrier of X
bool the carrier of X is non empty set
bool the carrier of R is non empty set
f0 " is Relation-like the carrier of X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of X, the carrier of R:]
[: the carrier of X, the carrier of R:] is Relation-like non empty set
bool [: the carrier of X, the carrier of R:] is non empty set
tcR is non empty add-closed left-ideal right-ideal Element of bool the carrier of X
(f0 ") .: tcR is non empty Element of bool the carrier of R
cR is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
S is non empty finite Element of bool the carrier of R
S -Ideal is non empty add-closed left-ideal right-ideal finitely_generated Element of bool the carrier of R
f0 " is Relation-like Function-like set
f0 .: ((f0 ") .: tcR) is non empty Element of bool the carrier of X
f0 " tcR is Element of bool the carrier of R
f0 .: (f0 " tcR) is Element of bool the carrier of X
f0 .: S is non empty finite Element of bool the carrier of X
(f0 .: S) -Ideal is non empty add-closed left-ideal right-ideal finitely_generated Element of bool the carrier of X
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty non trivial set
Polynom-Ring (0,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative right-distributive right_unital well-unital left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (0,R)) is non empty set
[: the carrier of R, the carrier of (Polynom-Ring (0,R)):] is Relation-like non empty set
bool [: the carrier of R, the carrier of (Polynom-Ring (0,R)):] is non empty set
X is Relation-like Function-like set
dom X is set
f0 is set
rng X is set
tcR is set
X . tcR is set
[:(Bags {}), the carrier of R:] is Relation-like non empty set
bool [:(Bags {}), the carrier of R:] is non empty set
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> cR is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{cR} is non empty trivial finite 1 -element set
[:(Bags {}),{cR}:] is Relation-like non empty set
S is Relation-like Bags {} -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
Support S is functional Element of bool (Bags {})
bool (Bags {}) is non empty set
Bags 0 is functional non empty Element of bool (Bags 0)
Bags 0 is non empty set
bool (Bags 0) is non empty set
[:(Bags 0), the carrier of R:] is Relation-like non empty set
bool [:(Bags 0), the carrier of R:] is non empty set
tcR is Relation-like Bags {} -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
dom tcR is functional non empty Element of bool (Bags {})
rng tcR is non empty Element of bool the carrier of R
bool the carrier of R is non empty set
tcR . {} is set
{(tcR . {})} is non empty trivial finite 1 -element set
(Bags {}) --> (tcR . {}) is Relation-like Bags {} -defined {(tcR . {})} -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}),{(tcR . {})}:]
[:(Bags {}),{(tcR . {})}:] is Relation-like non empty set
bool [:(Bags {}),{(tcR . {})}:] is non empty set
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
X . cR is set
EmptyBag {} is Relation-like non-empty empty-yielding {} -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of Bags {}
{} --> 0 is Relation-like non-empty empty-yielding {} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one constant functional empty total quasi_total Function-yielding V22() ordinal T-Sequence-like natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of bool [:{},NAT:]
[:{},NAT:] is Relation-like RAT -valued INT -valued V62() V63() V64() V65() set
bool [:{},NAT:] is non empty set
{0} is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
[:{},{0}:] is Relation-like RAT -valued INT -valued finite V62() V63() V64() V65() set
f0 is Relation-like the carrier of R -defined the carrier of (Polynom-Ring (0,R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of (Polynom-Ring (0,R)):]
tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
tcR + cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . (tcR + cR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
S is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
tcPR is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
tcPR . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> cR is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{cR} is non empty trivial finite 1 -element set
[:(Bags {}),{cR}:] is Relation-like non empty set
((Bags {}) --> cR) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
S . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> tcR is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{tcR} is non empty trivial finite 1 -element set
[:(Bags {}),{tcR}:] is Relation-like non empty set
((Bags {}) --> tcR) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0X is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
0X . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> (tcR + cR) is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{(tcR + cR)} is non empty trivial finite 1 -element set
[:(Bags {}),{(tcR + cR)}:] is Relation-like non empty set
((Bags {}) --> (tcR + cR)) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(S . S) + (tcPR . S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S + tcPR is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
(f0 . tcR) + (f0 . cR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . tcR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
tcR * cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f0 . (tcR * cR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
<*(tcR * cR)*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
S is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
C is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
0X is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
0X . C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> (tcR * cR) is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{(tcR * cR)} is non empty trivial finite 1 -element set
[:(Bags {}),{(tcR * cR)}:] is Relation-like non empty set
((Bags {}) --> (tcR * cR)) . C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
decomp C is Relation-like NAT -defined K239(2,(Bags 0)) -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of K239(2,(Bags 0))
K239(2,(Bags 0)) is M10( Bags 0)
len CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (decomp C) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
CN is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
S . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> tcR is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{tcR} is non empty trivial finite 1 -element set
[:(Bags {}),{tcR}:] is Relation-like non empty set
((Bags {}) --> tcR) . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcPR is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
mm is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
tcPR . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Bags {}) --> cR is Relation-like Bags {} -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}), the carrier of R:]
{cR} is non empty trivial finite 1 -element set
[:(Bags {}),{cR}:] is Relation-like non empty set
((Bags {}) --> cR) . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom CN is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (decomp C) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
(decomp C) /. cn is Relation-like NAT -defined Bags 0 -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags 0))
(decomp C) . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
m1 is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
m2 is Relation-like non-empty empty-yielding 0 -defined RAT -valued Function-like one-to-one constant functional empty total Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() set
<*m1,m2*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
CN /. cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN . 1 is set
S . m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcPR . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(S . m1) * (tcPR . m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(decomp C) /. c is Relation-like NAT -defined Bags 0 -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags 0))
CN /. c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S *' tcPR is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
(f0 . tcR) * (f0 . cR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
[:{{}}, the carrier of R:] is Relation-like non empty set
bool [:{{}}, the carrier of R:] is non empty set
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{{}} --> (0. R) is Relation-like {{}} -defined the carrier of R -valued Function-like constant non empty total quasi_total finite finite-support Element of bool [:{{}}, the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:{{}},{(0. R)}:] is Relation-like non empty finite set
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{{}} --> (1_ R) is Relation-like {{}} -defined the carrier of R -valued Function-like constant non empty total quasi_total finite finite-support Element of bool [:{{}}, the carrier of R:]
{(1_ R)} is non empty trivial finite 1 -element set
[:{{}},{(1_ R)}:] is Relation-like non empty finite set
cR is Relation-like {{}} -defined the carrier of R -valued Function-like non empty total quasi_total finite finite-support Element of bool [:{{}}, the carrier of R:]
dom cR is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool {{}}
bool {{}} is non empty finite V34() set
tcR is Relation-like {{}} -defined the carrier of R -valued Function-like non empty total quasi_total finite finite-support Element of bool [:{{}}, the carrier of R:]
dom tcR is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool {{}}
dom ({{}} --> (0. R)) is functional non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool {{}}
1_ (Polynom-Ring (0,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
K409((Polynom-Ring (0,R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
the OneF of (Polynom-Ring (0,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
1_ (0,R) is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
0_ (0,R) is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags 0), the carrier of R:]
(0_ (0,R)) +* ({},(1_ R)) is Relation-like Bags 0 -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags 0), the carrier of R:]
({{}} --> (0. R)) +* ({},(1_ R)) is Relation-like {{}} -defined the carrier of R -valued Function-like non empty total quasi_total finite finite-support Element of bool [:{{}}, the carrier of R:]
{} .--> (1_ R) is Relation-like {{}} -defined the carrier of R -valued Function-like one-to-one finite finite-support set
{{}} --> (1_ R) is Relation-like {{}} -defined the carrier of R -valued {(1_ R)} -valued Function-like constant non empty total quasi_total finite finite-support Element of bool [:{{}},{(1_ R)}:]
bool [:{{}},{(1_ R)}:] is non empty finite V34() set
({{}} --> (0. R)) +* ({} .--> (1_ R)) is Relation-like Function-like finite finite-support set
f0 . (1_ R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (0,R))
S is set
dom f0 is non empty Element of bool the carrier of R
tcPR is set
f0 . S is set
f0 . tcPR is set
(Bags {}) --> S is Relation-like Bags {} -defined {S} -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}),{S}:]
{S} is non empty trivial finite 1 -element set
[:(Bags {}),{S}:] is Relation-like non empty set
bool [:(Bags {}),{S}:] is non empty set
{tcPR} is non empty trivial finite 1 -element set
(Bags {}) --> tcPR is Relation-like Bags {} -defined {tcPR} -valued Function-like constant non empty total quasi_total Element of bool [:(Bags {}),{tcPR}:]
[:(Bags {}),{tcPR}:] is Relation-like non empty set
bool [:(Bags {}),{tcPR}:] is non empty set
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty non trivial set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict add-associative right_zeroed V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (X,R)) is non empty set
[: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is non empty set
f0 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
tcR is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
tcR . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
Sum cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
C is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
C . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is Relation-like the carrier of (Polynom-Ring (X,R)) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:]
S * cR is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (S * cR) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
c . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S . C is set
C . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
S . C is set
C . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
len CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
c . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S * CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (S * CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn ^ cn is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
rng cn is finite set
rng CN is finite Element of bool the carrier of (Polynom-Ring (X,R))
bool the carrier of (Polynom-Ring (X,R)) is non empty set
rng cn is finite set
mm is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
Sum mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Sum mm) + (Sum CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S * CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
S * mm is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
S * CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
(S * mm) ^ (S * CN) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
dom CN is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
CN . 1 is set
rng CN is finite Element of bool the carrier of (Polynom-Ring (X,R))
m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
<*m2*> is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring (X,R)) *
the carrier of (Polynom-Ring (X,R)) * is functional non empty FinSequence-membered M10( the carrier of (Polynom-Ring (X,R)))
S . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*(S . m2)*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
lc is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
lc . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*(lc . f0)*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
Sum (S * CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m1 + lc is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m1 . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum (S * mm) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Sum (S * mm)) + (Sum (S * CN)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum (S * CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
len CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
c . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S * CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (S * CN) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
len C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S * C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (S * C) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*> the carrier of (Polynom-Ring (X,R)) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of (Polynom-Ring (X,R)) *
the carrier of (Polynom-Ring (X,R)) * is functional non empty FinSequence-membered M10( the carrier of (Polynom-Ring (X,R)))
<*> the carrier of R is Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
len cR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
len C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Sum C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
CN . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S * C is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (S * C) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
[: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is non empty set
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
the carrier of R is non empty non trivial set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
C is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
{ b₁ where b₁ is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1) : ( b₁ . X = a₁ & ( for b₂ being Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:] holds
( not b₂ = C . a₁ or b₁ | X in Support b₂ ) ) ) } is set
CN is set
C . CN is set
{ b₁ where b₁ is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1) : ( b₁ . X = CN & ( for b₂ being Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:] holds
( not b₂ = C . CN or b₁ | X in Support b₂ ) ) ) } is set
cn is set
cn is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
cn . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
bool (Bags (X + 1)) is non empty set
[:NAT,(bool (Bags (X + 1))):] is Relation-like non empty non trivial non finite set
bool [:NAT,(bool (Bags (X + 1))):] is non empty non trivial non finite set
CN is Relation-like NAT -defined bool (Bags (X + 1)) -valued Function-like non empty total quasi_total Element of bool [:NAT,(bool (Bags (X + 1))):]
c is set
len C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN .: (len C) is finite Element of bool (bool (Bags (X + 1)))
bool (bool (Bags (X + 1))) is non empty set
cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN . cn is functional Element of bool (Bags (X + 1))
C . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
{ b₁ where b₁ is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1) : ( b₁ . X = cn & ( for b₂ being Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:] holds
( not b₂ = C . cn or b₁ | X in Support b₂ ) ) ) } is set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Support cn is functional finite Element of bool (Bags X)
bool (Bags X) is non empty set
CN is set
mm is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
mm . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
mm | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Support cn is functional finite Element of bool (Bags X)
bool (Bags X) is non empty set
CN is set
mm is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,cn,mm) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
(X,cn,mm) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,cn,mm) | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Support m1 is functional finite Element of bool (Bags X)
m1 is non empty set
m2 is non empty set
lc is Element of m1
ev is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
ev . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
ev | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
E is Element of m2
P is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
P | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
[:m1,m2:] is Relation-like non empty set
bool [:m1,m2:] is non empty set
lc is Relation-like m1 -defined m2 -valued Function-like non empty total quasi_total Element of bool [:m1,m2:]
dom lc is non empty Element of bool m1
bool m1 is non empty set
ev is set
E is set
lc . ev is set
lc . E is set
LC is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
LC . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
P is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,cn,P) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
LC is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
LC . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
rng lc is non empty Element of bool m2
bool m2 is non empty set
card m1 is non empty ordinal cardinal set
card m2 is non empty ordinal cardinal set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Support cn is functional finite Element of bool (Bags X)
bool (Bags X) is non empty set
union (CN .: (len C)) is set
c is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
c . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (c . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
c | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
cn is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN . (c | X) is set
c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags (X + 1)), the carrier of R:]
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags (X + 1)), the carrier of R:]
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags (X + 1)), the carrier of R:]
Support cn is functional Element of bool (Bags (X + 1))
CN is set
dom CN is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
mm is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
mm . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (mm . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
mm | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cn . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m2 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
m1 . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
{ b₁ where b₁ is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT : not len C <= b₁ } is set
CN . (mm . X) is functional Element of bool (Bags (X + 1))
{ b₁ where b₁ is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1) : ( b₁ . X = mm . X & ( for b₂ being Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:] holds
( not b₂ = C . (mm . X) or b₁ | X in Support b₂ ) ) ) } is set
lc is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
Support lc is functional finite Element of bool (Bags X)
bool (Bags X) is non empty set
lc is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
mm is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
m2 is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
lc . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
mm . (lc . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
m2 . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
m1 . (lc | X) is set
S is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
C is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
C . CN is set
c is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
cn is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
cn . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
c . (cn | X) is set
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
C . CN is set
c is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
cn is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
cn . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
c . (cn | X) is set
f0 is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
tcR is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
0X is set
f0 . 0X is set
tcR . 0X is set
c is set
S is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
cn . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
cn | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
C is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
C . c is set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
cn . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN . c is set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
(R,X) is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
[: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is non empty set
cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
cR + S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
(R,X) . (cR + S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . cR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
((R,X) . cR) + ((R,X) . S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
cR + S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
the carrier of R is non empty non trivial set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
(R,X) . (cR + S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
S is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
tcPR is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
0X is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
tcPR + 0X is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
cn is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(tcPR + 0X) . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
tcPR . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
0X . (cn . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
cn | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
mm is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m2 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
mm . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m1 . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
c . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(tcPR . (cn . X)) + (0X . (cn . X)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
mm + m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
C is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
C . cn is set
(mm + m1) . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(CN . cn) + (c . cn) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN + c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
(CN + c) . cn is set
((R,X) . cR) + ((R,X) . S) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
the carrier of R is non empty non trivial set
the carrier of (Polynom-Ring (X,R)) is non empty set
tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
tcPR * 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
(R,X) . (tcPR * 0X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
((R,X) . tcPR) * ((R,X) . 0X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
tcPR * 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
(R,X) . (tcPR * 0X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
((R,X) . tcPR) * ((R,X) . 0X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
CN is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
S is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
C is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
S *' C is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
cn is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
cn *' CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
mm is set
m1 is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
m1 . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(m1 . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
CN . (m1 . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
m2 is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring (X,R))
len m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
dom m2 is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
Bags X is non empty set
Bags X is functional non empty Element of bool (Bags X)
bool (Bags X) is non empty set
m1 | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
lc is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
decomp lc is Relation-like NAT -defined K239(2,(Bags X)) -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of K239(2,(Bags X))
K239(2,(Bags X)) is M10( Bags X)
len (decomp lc) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
cn . m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
decomp m1 is Relation-like NAT -defined K239(2,(Bags (X + 1))) -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of K239(2,(Bags (X + 1)))
K239(2,(Bags (X + 1))) is M10( Bags (X + 1))
len (decomp m1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
E is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len E is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom E is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
P is set
LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (LC -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
((m1 . X) + 1) -' LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' LC) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
LC is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
i is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
LC *' i is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(LC *' i) . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
y is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum y is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len y is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom y is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
n is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
v -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (v -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
v is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
((m1 . X) + 1) -' v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' v) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
u is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
n *' v is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(n *' v) . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum u is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
a is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom u is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(decomp lc) /. a is Relation-like NAT -defined Bags X -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags X))
u /. a is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[:(dom m2),( the carrier of R *):] is Relation-like set
bool [:(dom m2),( the carrier of R *):] is non empty set
P is Relation-like dom m2 -defined the carrier of R * -valued Function-like total quasi_total Function-yielding V22() finite finite-support Element of bool [:(dom m2),( the carrier of R *):]
rng P is functional finite FinSequence-membered Element of bool ( the carrier of R *)
bool ( the carrier of R *) is non empty set
dom P is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (dom m2)
bool (dom m2) is non empty finite V34() set
LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
Seg LC is finite LC -element V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
[: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is non empty set
ev is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
ev . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
LC is Relation-like the carrier of (Polynom-Ring (X,R)) -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:]
LC * m2 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum (LC * m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
c . m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
LC is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
dom LC is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
1 -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (1 -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
((m1 . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
LC /. 1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
LC . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
i is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
y is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
len (LC . 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
divisors m1 is Relation-like NAT -defined Bags (X + 1) -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support FinSequence of Bags (X + 1)
divisors lc is Relation-like NAT -defined Bags X -valued Function-like one-to-one non empty Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support FinSequence of Bags X
dom (divisors lc) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
FlattenSeq LC is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
y is set
dom (decomp m1) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
dom (divisors m1) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors m1) . n is Relation-like Function-like set
rng (divisors m1) is functional non empty finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
v is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
v . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(v . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
v | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
((v . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (((v . X) + 1) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
((m1 . X) + 1) -' ((v . X) + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' ((v . X) + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
v is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
a is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
u is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
rng (divisors lc) is functional non empty finite Element of bool (Bags X)
bool (Bags X) is non empty set
yb2n is set
(divisors lc) . yb2n is Relation-like Function-like set
LC | (((v . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (((v . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
Sum (Card (LC | (((v . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
b1n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (((v . X) + 1) -' 1)))) + b1n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
1 + 0 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
LC . ((v . X) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len (LC . ((v . X) + 1)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (LC . ((v . X) + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (decomp lc) is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
dom (FlattenSeq LC) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
sb1n1 is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(divisors m1) . i is Relation-like Function-like set
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors lc) . i is Relation-like Function-like set
sb1n1 | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
sb1n1 . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(sb1n1 . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
LC . ((sb1n1 . X) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (LC . ((sb1n1 . X) + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
((sb1n1 . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | (((sb1n1 . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (((sb1n1 . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
Sum (Card (LC | (((sb1n1 . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (((sb1n1 . X) + 1) -' 1)))) + i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
[:(dom E),(dom (FlattenSeq LC)):] is Relation-like RAT -valued INT -valued finite V62() V63() V64() V65() set
bool [:(dom E),(dom (FlattenSeq LC)):] is non empty finite V34() set
y is Relation-like dom E -defined dom (FlattenSeq LC) -valued Function-like quasi_total finite V62() V63() V64() V65() finite-support Element of bool [:(dom E),(dom (FlattenSeq LC)):]
Card LC is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
dom (Card LC) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(Card LC) . 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum (Card LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (FlattenSeq LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom y is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (dom E)
bool (dom E) is non empty finite V34() set
n is set
v is set
y . n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
y . v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors m1) . u is Relation-like Function-like set
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors m1) . v is Relation-like Function-like set
a is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
a | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
yb2n is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
yb2n | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
a . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(a . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((a . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Card LC) | (((a . X) + 1) -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
LC | (((a . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (((a . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
yb2n . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(yb2n . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((yb2n . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Card LC) | (((yb2n . X) + 1) -' 1) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
LC | (((yb2n . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (((yb2n . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
b2n is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
i is set
(divisors lc) . i is Relation-like Function-like set
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC . ((yb2n . X) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (LC . ((yb2n . X) + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
b1n is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
sb1n1 is set
(divisors lc) . sb1n1 is Relation-like Function-like set
y . u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum (Card (LC | (((a . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
B1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (((a . X) + 1) -' 1)))) + B1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
y . v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum (Card (LC | (((yb2n . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (((yb2n . X) + 1) -' 1)))) + i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC . ((a . X) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (LC . ((a . X) + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(X,(a . X),b1n) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
rng y is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (dom (FlattenSeq LC))
bool (dom (FlattenSeq LC)) is non empty finite V34() set
n is set
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(FlattenSeq LC) . v is set
u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC . u is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (LC . u) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
u -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | (u -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (u -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
Sum (Card (LC | (u -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (u -' 1)))) + v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(LC . u) . v is set
(divisors lc) /. v is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
a is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(X,(u -' 1),a) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
S . (u -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
((m1 . X) + 1) -' u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' u) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(X,(u -' 1),a) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((X,(u -' 1),a) . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
b1n is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
b2n is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
len (LC . u) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors lc) . v is Relation-like Function-like set
i is set
m1 . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
lc . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,(u -' 1),a) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,(u -' 1),a) | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
((X,(u -' 1),a) | X) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
a . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
u - 1 is V42() V43() V44() ext-real set
((m1 . X) + 1) - 1 is V42() V43() V44() ext-real set
succ X is set
{X} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
X \/ {X} is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() set
(X,(u -' 1),a) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,(u -' 1),a) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,(u -' 1),a) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
i is set
dom (X,(u -' 1),a) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool (X + 1)
bool (X + 1) is non empty finite V34() set
(X,(u -' 1),a) . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
i is set
i is set
(divisors m1) . i is Relation-like Function-like set
(X,(u -' 1),a) | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
y . i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(((X,(u -' 1),a) . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | ((((X,(u -' 1),a) . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | ((((X,(u -' 1),a) . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
Sum (Card (LC | ((((X,(u -' 1),a) . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | ((((X,(u -' 1),a) . X) + 1) -' 1)))) + v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum LC is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len (Sum LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (Sum LC) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
n -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (n -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
((m1 . X) + 1) -' n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (((m1 . X) + 1) -' n) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
v is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
u is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
v *' u is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
LC . n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
a is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
m2 . n is set
(S . (n -' 1)) * (C . (((m1 . X) + 1) -' n)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Sum LC) /. n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Sum LC) . n is set
LC /. n is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
yb2n is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
Sum yb2n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(v *' u) . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
LC . (m2 . n) is set
(LC * m2) . n is set
n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(divisors m1) /. n is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
(decomp m1) /. n is Relation-like NAT -defined Bags (X + 1) -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags (X + 1)))
E /. n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
u is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
v is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
<*u,v*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
cn . u is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN . v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cn . u) * (CN . v) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
v is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
m1 -' v is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
<*v,(m1 -' v)*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
u . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (u . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
v . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C . (v . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
u | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
v | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
E . n is set
a is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
b1n is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
a . b1n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(a . b1n) * (CN . v) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
yb2n is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
b2n is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
yb2n . b2n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(a . b1n) * (yb2n . b2n) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(divisors m1) . n is Relation-like Function-like set
i is set
(divisors lc) . i is Relation-like Function-like set
(u . X) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((m1 . X) + 1) -' ((u . X) + 1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((m1 . X) + 1) - ((u . X) + 1) is V42() V43() V44() ext-real set
(m1 . X) - (u . X) is V42() V43() V44() ext-real set
((m1 . X) - (u . X)) + 1 is V42() V43() V44() ext-real set
(((m1 . X) - (u . X)) + 1) - 1 is V42() V43() V44() ext-real set
(m1 . X) -' (u . X) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((u . X) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((u . X) + 1) - 1 is V42() V43() V44() ext-real set
S . (((u . X) + 1) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
LC . ((u . X) + 1) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom (LC . ((u . X) + 1)) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(decomp lc) /. i is Relation-like NAT -defined Bags X -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like finite-support Element of K239(2,(Bags X))
sb1n1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
sb1n1 /. i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
B1 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
B2 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
<*B1,B2*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
a . B1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
yb2n . B2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(a . B1) * (yb2n . B2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
y . n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC | (((u . X) + 1) -' 1) is Relation-like NAT -defined the carrier of R * -valued Function-like Function-yielding V22() finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of R *
Card (LC | (((u . X) + 1) -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V62() V63() V64() V65() V257() finite-support FinSequence of NAT
Sum (Card (LC | (((u . X) + 1) -' 1))) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(Sum (Card (LC | (((u . X) + 1) -' 1)))) + i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(FlattenSeq LC) . (y . n) is set
(LC . ((u . X) + 1)) . i is set
(divisors lc) /. i is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
B19 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
lc -' B19 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
<*B19,(lc -' B19)*> is Relation-like NAT -defined Function-like non empty finite 2 -element FinSequence-like FinSubsequence-like finite-support set
C . (((m1 . X) + 1) -' ((u . X) + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
len sb1n1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom sb1n1 is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (LC * m2) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
Sum (FlattenSeq LC) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
card (dom E) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
card y is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
card (dom (FlattenSeq LC)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
[:(dom E),(dom E):] is Relation-like RAT -valued INT -valued finite V62() V63() V64() V65() set
bool [:(dom E),(dom E):] is non empty finite V34() set
c . mm is set
cn . mm is set
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
1_ (Polynom-Ring (Polynom-Ring (X,R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
K409((Polynom-Ring (Polynom-Ring (X,R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
the OneF of (Polynom-Ring (Polynom-Ring (X,R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
the carrier of R is non empty non trivial set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
1_ (Polynom-Ring ((X + 1),R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
K409((Polynom-Ring ((X + 1),R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
the OneF of (Polynom-Ring ((X + 1),R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . (1_ (Polynom-Ring (Polynom-Ring (X,R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
0X is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
(R,X) . 0X is set
CN is set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
c is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
c . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X . (c . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
C is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
C . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
c | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
cn . (c | X) is set
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,0,(EmptyBag X)) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
EmptyBag (X + 1) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
1_. (Polynom-Ring (X,R)) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
0_. (Polynom-Ring (X,R)) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
NAT --> (0. (Polynom-Ring (X,R))) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
{(0. (Polynom-Ring (X,R)))} is non empty trivial finite 1 -element set
[:NAT,{(0. (Polynom-Ring (X,R)))}:] is Relation-like non empty non trivial non finite set
K409((Polynom-Ring (X,R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the OneF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(0_. (Polynom-Ring (X,R))) +* (0,K409((Polynom-Ring (X,R)))) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
(1_. (Polynom-Ring (X,R))) . 0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
1_ (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
1_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
1_ ((X + 1),R) is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
(1_ ((X + 1),R)) . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
S . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
1_. (Polynom-Ring (X,R)) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
0_. (Polynom-Ring (X,R)) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
NAT --> (0. (Polynom-Ring (X,R))) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
{(0. (Polynom-Ring (X,R)))} is non empty trivial finite 1 -element set
[:NAT,{(0. (Polynom-Ring (X,R)))}:] is Relation-like non empty non trivial non finite set
K409((Polynom-Ring (X,R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the OneF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(0_. (Polynom-Ring (X,R))) +* (0,K409((Polynom-Ring (X,R)))) is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
(1_. (Polynom-Ring (X,R))) . (c . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
1_ (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
1_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,0,(EmptyBag X)) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
EmptyBag (X + 1) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
1_ ((X + 1),R) is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
(1_ ((X + 1),R)) . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
S . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
S is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
S . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
S . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C . CN is set
S . CN is set
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
the carrier of R is non empty non trivial set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
(R,X) . tcPR is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . S is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
CN is set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
0X is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
c is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
S . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
CN is set
mm is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(X,c,mm) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
(X,c,mm) | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
(X,c,mm) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
cn . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
C . (X,c,mm) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
cn . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn . CN is set
cn . CN is set
0X . CN is set
S . CN is set
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
[: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
the carrier of R is non empty non trivial set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
c is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional Element of bool (Bags X)
bool (Bags X) is non empty set
mm is set
m1 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
CN . m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,c,m1) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN . (X,c,m1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional Element of bool (Bags X)
bool (Bags X) is non empty set
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional Element of bool (Bags X)
bool (Bags X) is non empty set
mm is functional non empty Element of bool (Bags X)
m2 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of mm
CN . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,c,m2) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN . (X,c,m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m1 is functional non empty Element of bool (Bags (X + 1))
[:mm,m1:] is Relation-like non empty set
bool [:mm,m1:] is non empty set
m2 is Relation-like mm -defined m1 -valued Function-like non empty total quasi_total Function-yielding V22() Element of bool [:mm,m1:]
dom m2 is functional non empty Element of bool mm
bool mm is non empty set
lc is set
ev is set
m2 . lc is Relation-like Function-like set
m2 . ev is Relation-like Function-like set
E is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,c,E) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
P is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,c,P) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
(X,c,E) | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
rng m2 is functional non empty Element of bool m1
bool m1 is non empty set
card mm is non empty ordinal cardinal set
card m1 is non empty ordinal cardinal set
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional Element of bool (Bags X)
bool (Bags X) is non empty set
CN is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
Support CN is functional Element of bool (Bags X)
bool (Bags X) is non empty set
mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
lc is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
m2 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
m2 . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,c,lc) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN . (X,c,lc) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m2 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
lc is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
m2 . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,c,lc) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
CN . (X,c,lc) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
m1 is set
m2 is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,cn,m2) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
CN . (X,cn,m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
mm is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
mm . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
mm . m1 is set
(Bags X) --> (0. R) is Relation-like Bags X -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:(Bags X),{(0. R)}:] is Relation-like non empty set
((Bags X) --> (0. R)) . m1 is set
dom ((Bags X) --> (0. R)) is functional non empty Element of bool (Bags X)
dom mm is functional non empty Element of bool (Bags X)
cn . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
[:(Bags (X + 1)),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite V62() V63() V64() V65() set
bool [:(Bags (X + 1)),NAT:] is non empty non trivial non finite set
CN is Relation-like Bags (X + 1) -defined NAT -valued Function-like non empty total quasi_total V62() V63() V64() V65() Element of bool [:(Bags (X + 1)),NAT:]
cn is functional non empty finite Element of bool (Bags (X + 1))
CN .: cn is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
max (CN .: cn) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() set
m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
mm + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
ev is set
E is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(X,m1,E) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
CN . (X,m1,E) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(X,m1,E) . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN . (X,m1,E) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
lc . E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc . ev is set
(Bags X) --> (0. R) is Relation-like Bags X -defined the carrier of R -valued Function-like constant non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:(Bags X),{(0. R)}:] is Relation-like non empty set
((Bags X) --> (0. R)) . ev is set
dom ((Bags X) --> (0. R)) is functional non empty Element of bool (Bags X)
dom lc is functional non empty Element of bool (Bags X)
cn . m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0. (Polynom-Ring (X,R)) is V102( Polynom-Ring (X,R)) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
the ZeroF of (Polynom-Ring (X,R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
Support CN is functional finite Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
mm is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
m2 is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m2 . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
ev is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
m1 . ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,lc,ev) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
mm . (X,lc,ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
C is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
CN is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
CN . c is set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
mm is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,CN,mm) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
c . (X,CN,mm) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
c is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
CN . c is set
cn is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
mm is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
cn . mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,CN,mm) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
c . (X,CN,mm) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
tcPR is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
0X is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
S is set
C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
tcPR . C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
0X . C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
cn is set
c is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
cn is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
cn . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
m2 is set
mm is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
mm . m2 is set
CN is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
lc is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(X,CN,lc) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
CN . (X,CN,lc) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m1 is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
m1 . m2 is set
c . cn is set
cn . cn is set
tcPR . S is set
0X . S is set
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
(R,X) is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like one-to-one non empty total quasi_total unity-preserving multiplicative additive Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
[: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is non empty set
(R,X) is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
[: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is non empty set
f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . f0 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
(R,X) . ((R,X) . f0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
Bags (X + 1) is non empty set
bool (Bags (X + 1)) is non empty set
the carrier of R is non empty non trivial set
[:(Bags (X + 1)), the carrier of R:] is Relation-like non empty set
bool [:(Bags (X + 1)), the carrier of R:] is non empty set
the carrier of (Polynom-Ring (X,R)) is non empty set
[:NAT, the carrier of (Polynom-Ring (X,R)):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring (X,R)):] is non empty non trivial non finite set
0X is set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
tcPR is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of (Polynom-Ring (X,R)):]
S is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
S . X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . (S . X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
S | X is Relation-like X -defined X + 1 -defined RAT -valued Function-like finite V62() V63() V64() V65() finite-support set
CN is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(X,(S . X),CN) is Relation-like X + 1 -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags (X + 1)
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Bags (X + 1) is non empty set
Bags (X + 1) is functional non empty Element of bool (Bags (X + 1))
bool (Bags (X + 1)) is non empty set
S is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
S . 0X is set
C is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
C . CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cR is Relation-like Bags (X + 1) -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags (X + 1)), the carrier of R:]
cR . 0X is set
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
[: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is non empty set
(R,X) is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like one-to-one non empty total quasi_total unity-preserving multiplicative additive Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
tcPR is set
dom (R,X) is non empty Element of bool the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
bool the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
(R,X) is Relation-like the carrier of (Polynom-Ring ((X + 1),R)) -defined the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):]
[: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring ((X + 1),R)), the carrier of (Polynom-Ring (Polynom-Ring (X,R))):] is non empty set
0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
(R,X) . 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (Polynom-Ring (X,R)))
(R,X) . ((R,X) . 0X) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring ((X + 1),R))
rng (R,X) is non empty Element of bool the carrier of (Polynom-Ring ((X + 1),R))
bool the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() Noetherian doubleLoopStr
Polynom-Ring R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty set
the carrier of (Polynom-Ring R) is non empty set
bool the carrier of (Polynom-Ring R) is non empty set
cR is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
bool cR is non empty set
tcPR is Element of bool the carrier of (Polynom-Ring R)
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR is Element of bool the carrier of (Polynom-Ring R)
0X is non empty Element of bool the carrier of (Polynom-Ring R)
0X -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (0X -Ideal) is Element of bool the carrier of (Polynom-Ring R)
cR -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
C is non empty Element of bool the carrier of (Polynom-Ring R)
(R,C) is non empty Element of bool C
bool C is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
{ b₁ where b₁ is Element of C : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in C or len b₂ <= len b₃ ) } is set
CN is set
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{c} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
0X \/ {c} is non empty Element of bool the carrier of (Polynom-Ring R)
cn is non empty Element of bool the carrier of (Polynom-Ring R)
cn is non empty Element of bool the carrier of (Polynom-Ring R)
cn -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (cn -Ideal) is Element of bool the carrier of (Polynom-Ring R)
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(R,cn) is non empty Element of bool cn
bool cn is non empty set
{ b₁ where b₁ is Element of cn : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in cn or len b₂ <= len b₃ ) } is set
{CN} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
tcPR \/ {CN} is non empty set
tcPR is Element of bool the carrier of (Polynom-Ring R)
[:NAT,(bool the carrier of (Polynom-Ring R)):] is Relation-like non empty non trivial non finite set
bool [:NAT,(bool the carrier of (Polynom-Ring R)):] is non empty non trivial non finite set
0. (Polynom-Ring R) is V102( Polynom-Ring R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
the ZeroF of (Polynom-Ring R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{(0. (Polynom-Ring R))} is non empty trivial finite 1 -element add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
S is Relation-like NAT -defined bool the carrier of (Polynom-Ring R) -valued Function-like non empty total quasi_total Element of bool [:NAT,(bool the carrier of (Polynom-Ring R)):]
S . 0 is Element of bool the carrier of (Polynom-Ring R)
tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is Element of bool the carrier of (Polynom-Ring R)
0X is non empty finite Element of bool cR
tcPR + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (tcPR + 1) is Element of bool the carrier of (Polynom-Ring R)
S is non empty Element of bool the carrier of (Polynom-Ring R)
C is non empty Element of bool the carrier of (Polynom-Ring R)
S -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (S -Ideal) is Element of bool the carrier of (Polynom-Ring R)
(R,C) is non empty Element of bool C
bool C is non empty set
[:NAT, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
{ b₁ where b₁ is Element of C : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in C or len b₂ <= len b₃ ) } is set
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{CN} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
0X \/ {CN} is non empty finite set
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{CN} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
0X \/ {CN} is non empty finite set
tcPR is set
tcPR is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . tcPR is Element of bool the carrier of (Polynom-Ring R)
tcPR + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (tcPR + 1) is Element of bool the carrier of (Polynom-Ring R)
0X is non empty Element of bool the carrier of (Polynom-Ring R)
S is non empty Element of bool the carrier of (Polynom-Ring R)
0X -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (0X -Ideal) is Element of bool the carrier of (Polynom-Ring R)
C is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(R,S) is non empty Element of bool S
bool S is non empty set
{ b₁ where b₁ is Element of S : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in S or len b₂ <= len b₃ ) } is set
{C} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
(S . tcPR) \/ {C} is non empty Element of bool the carrier of (Polynom-Ring R)
[:NAT, the carrier of (Polynom-Ring R):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring R):] is non empty non trivial non finite set
tcPR is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of (Polynom-Ring R):]
0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . 0X is Element of bool the carrier of (Polynom-Ring R)
S . S is Element of bool the carrier of (Polynom-Ring R)
C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
0X + C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0X + CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (0X + CN) is Element of bool the carrier of (Polynom-Ring R)
CN + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
0X + (CN + 1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (0X + (CN + 1)) is Element of bool the carrier of (Polynom-Ring R)
(0X + CN) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . ((0X + CN) + 1) is Element of bool the carrier of (Polynom-Ring R)
c is non empty Element of bool the carrier of (Polynom-Ring R)
cn is non empty Element of bool the carrier of (Polynom-Ring R)
c -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (c -Ideal) is Element of bool the carrier of (Polynom-Ring R)
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(R,cn) is non empty Element of bool cn
bool cn is non empty set
{ b₁ where b₁ is Element of cn : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in cn or len b₂ <= len b₃ ) } is set
{cn} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
(S . (0X + CN)) \/ {cn} is non empty Element of bool the carrier of (Polynom-Ring R)
0X + 0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (0X + 0) is Element of bool the carrier of (Polynom-Ring R)
0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{(tcPR . 0X)} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
C is non empty Element of bool the carrier of (Polynom-Ring R)
S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . S is Element of bool the carrier of (Polynom-Ring R)
CN is non empty Element of bool the carrier of (Polynom-Ring R)
C -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (C -Ideal) is Element of bool the carrier of (Polynom-Ring R)
S + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (S + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . S) \/ {(tcPR . 0X)} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,CN) is non empty Element of bool CN
bool CN is non empty set
{ b₁ where b₁ is Element of CN : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in CN or len b₂ <= len b₃ ) } is set
S is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
0_. R is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
NAT --> (0. R) is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:NAT, the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:NAT,{(0. R)}:] is Relation-like non empty non trivial non finite set
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(len S) - 1 is V42() V43() V44() ext-real set
0X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . 0X is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
S is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len S is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len S) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . ((len S) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
0X is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
bool the carrier of R is non empty set
C is non empty Element of bool the carrier of R
C -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
S is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of R:]
C is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
C + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (C + 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
S | (C + 1) is Relation-like NAT -defined C + 1 -defined NAT -defined the carrier of R -valued Function-like finite finite-support Element of bool [:NAT, the carrier of R:]
rng (S | (C + 1)) is finite Element of bool the carrier of R
(rng (S | (C + 1))) -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
tcPR . (C + 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
Segm (C + 1) is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
S | (Segm (C + 1)) is Relation-like NAT -defined NAT -defined Segm (C + 1) -defined NAT -defined the carrier of R -valued Function-like non empty total Element of bool [:NAT, the carrier of R:]
rng (S | (Segm (C + 1))) is non empty Element of bool the carrier of R
CN is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len CN) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . (C + 1) is Element of bool the carrier of (Polynom-Ring R)
{(tcPR . (C + 1))} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
c is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
tcPR . cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
cn is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
CN is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative set
c + mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c + m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . (c + m1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
m2 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
{(tcPR . (c + m1))} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
E is non empty Element of bool the carrier of (Polynom-Ring R)
ev is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . ev is Element of bool the carrier of (Polynom-Ring R)
P is non empty Element of bool the carrier of (Polynom-Ring R)
E -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (E -Ideal) is Element of bool the carrier of (Polynom-Ring R)
ev + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (ev + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . ev) \/ {(tcPR . (c + m1))} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,P) is non empty Element of bool P
bool P is non empty set
{ b₁ where b₁ is Element of P : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in P or len b₂ <= len b₃ ) } is set
m1 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
c + (m1 + 1) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
tcPR . (c + (m1 + 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
{(tcPR . (c + (m1 + 1)))} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
LC is non empty Element of bool the carrier of (Polynom-Ring R)
LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . LC is Element of bool the carrier of (Polynom-Ring R)
i is non empty Element of bool the carrier of (Polynom-Ring R)
LC -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (LC -Ideal) is Element of bool the carrier of (Polynom-Ring R)
LC + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (LC + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . LC) \/ {(tcPR . (c + (m1 + 1)))} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,i) is non empty Element of bool i
bool i is non empty set
{ b₁ where b₁ is Element of i : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in i or len b₂ <= len b₃ ) } is set
(c + m1) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
lc is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m2 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c + 0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . (c + 0) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c is set
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn * cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cn * cn) * CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn * cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cn * cn) * CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom (S | (Segm (C + 1))) is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
mm is set
(S | (Segm (C + 1))) . mm is set
(R,0,CN) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
((len CN) -' 1) + 0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
ev is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
tcPR . ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
P is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len P is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len CN) -' (len P) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(R,((len CN) -' (len P)),cn) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
dom S is non empty V72() V73() V74() V75() V76() V77() V290() V292() Element of bool NAT
(dom S) /\ (Segm (C + 1)) is V72() V73() V74() V75() V76() V77() V292() Element of bool NAT
NAT /\ (Segm (C + 1)) is V72() V73() V74() V75() V76() V77() V292() Element of bool NAT
(len P) - (len P) is V42() V43() V44() ext-real set
(len CN) - (len P) is V42() V43() V44() ext-real set
len (R,((len CN) -' (len P)),cn) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len CN) - (len P)) + 1 is V42() V43() V44() ext-real set
(len P) - 1 is V42() V43() V44() ext-real set
(len (R,((len CN) -' (len P)),cn)) + ((len P) - 1) is V42() V43() V44() ext-real set
(len CN) - ((len P) - 1) is V42() V43() V44() ext-real set
((len CN) - ((len P) - 1)) + ((len P) - 1) is V42() V43() V44() ext-real set
(R,((len CN) -' (len P)),cn) *' P is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
len ((R,((len CN) -' (len P)),cn) *' P) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len (R,((len CN) -' (len P)),cn)) + (len P) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len (R,((len CN) -' (len P)),cn)) + (len P)) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
0 + 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V49() V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of NAT
((len P) - 1) + (len (R,((len CN) -' (len P)),cn)) is V42() V43() V44() ext-real set
((len (R,((len CN) -' (len P)),cn)) + (len P)) - 1 is V42() V43() V44() ext-real set
(len CN) - 1 is V42() V43() V44() ext-real set
((len CN) - (len P)) + ((len P) - 1) is V42() V43() V44() ext-real set
(len P) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len P) -' 1) + ((len CN) -' (len P)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC * (tcPR . ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(LC * (tcPR . ev)) * i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
((R,((len CN) -' (len P)),cn) *' P) *' (R,0,CN) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
S . ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((R,((len CN) -' (len P)),cn) *' P) . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
u is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
tcPR . v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
len u is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len u) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
u . ((len u) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
n is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
n . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(tcPR . ev)} is non empty trivial finite 1 -element Element of bool the carrier of (Polynom-Ring R)
ev + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (ev + 1) is Element of bool the carrier of (Polynom-Ring R)
u is non empty Element of bool the carrier of (Polynom-Ring R)
v is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . v is Element of bool the carrier of (Polynom-Ring R)
v is non empty Element of bool the carrier of (Polynom-Ring R)
u -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (u -Ideal) is Element of bool the carrier of (Polynom-Ring R)
v + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (v + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . v) \/ {(tcPR . ev)} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,v) is non empty Element of bool v
bool v is non empty set
{ b₁ where b₁ is Element of v : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in v or len b₂ <= len b₃ ) } is set
len (R,0,CN) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len ((R,((len CN) -' (len P)),cn) *' P)) + (len (R,0,CN)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len CN) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((len ((R,((len CN) -' (len P)),cn) *' P)) + (len (R,0,CN))) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len CN) + 1) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (((R,((len CN) -' (len P)),cn) *' P) *' (R,0,CN)) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
((len CN) + 1) - 1 is V42() V43() V44() ext-real set
len n is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len CN) + 0 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn * cn is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(cn * cn) * CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[: the carrier of R, the carrier of (Polynom-Ring R):] is Relation-like non empty set
bool [: the carrier of R, the carrier of (Polynom-Ring R):] is non empty set
c is Relation-like the carrier of R -defined the carrier of (Polynom-Ring R) -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of (Polynom-Ring R):]
CN is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
Sum CN is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
cn is non empty Element of bool the carrier of (Polynom-Ring R)
mm is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
c * mm is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
len mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom c is non empty Element of bool the carrier of R
rng mm is finite Element of bool the carrier of R
dom mm is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom (c * mm) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
len (c * mm) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m2 is set
dom (c * mm) is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
(c * mm) /. m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
mm /. m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
E is Element of rng (S | (Segm (C + 1)))
lc * E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(lc * E) * ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
mm . m2 is set
c . (mm /. m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
i is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
P is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
P * LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(P * LC) * LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
i . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(c * mm) . m2 is set
n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
y is Element of cn
n * y is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(n * y) * v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
m2 is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
mm is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
c * mm is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m1 + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
m2 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
len m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c * m2 is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom m2 is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
{1} is non empty trivial finite V34() 1 -element V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
m2 /. 1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
E is Element of rng (S | (Segm (C + 1)))
lc * E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(lc * E) * ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
m2 . 1 is set
c . (m2 . 1) is set
i is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
P is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
P * LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(P * LC) * LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
i . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
<*(m2 . 1)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
y is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
len y is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c . ((lc * E) * ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
<*(c . ((lc * E) * ev))*> is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring R) *
the carrier of (Polynom-Ring R) * is functional non empty FinSequence-membered M10( the carrier of (Polynom-Ring R))
Sum y is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
lc is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
len lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
ev is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*E*> is Relation-like NAT -defined the carrier of R -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
ev ^ <*E*> is Relation-like NAT -defined the carrier of R -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len <*E*> is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
c * <*E*> is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum <*E*> is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
P is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
LC is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
Sum P is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
LC . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len ev is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len ev) + (len <*E*>) is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
(len ev) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
c * ev is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
LC is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
i is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
Sum LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
i . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
i + LC is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
(Sum ev) + E is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(i . ((len CN) -' 1)) + (LC . ((len CN) -' 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(i + LC) . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len (i + LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (i + LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (i + LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
len (i + LC) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom c is non empty Element of bool the carrier of R
rng ev is finite Element of bool the carrier of R
rng <*E*> is non empty trivial finite 1 -element Element of bool the carrier of R
(rng ev) \/ (rng <*E*>) is non empty finite Element of bool the carrier of R
c * (ev ^ <*E*>) is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(Sum LC) + (Sum P) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
v is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
n ^ v is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
m1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of rng (S | (Segm (C + 1)))
len m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c * m1 is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
Sum m1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
<*> the carrier of R is Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of R *
the carrier of R * is functional non empty FinSequence-membered M10( the carrier of R)
m2 is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
len m2 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Sum m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
lc is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
lc . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
<*> the carrier of (Polynom-Ring R) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring R) -valued Function-like one-to-one constant functional empty Function-yielding V22() ordinal natural finite finite-yielding V34() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V42() V43() V44() ext-real non positive non negative V62() V63() V64() V65() V72() V73() V74() V75() V76() V77() V78() FinSequence-yielding finite-support V292() V293() V294() V295() Element of the carrier of (Polynom-Ring R) *
the carrier of (Polynom-Ring R) * is functional non empty FinSequence-membered M10( the carrier of (Polynom-Ring R))
m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
c * CN is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring R)
mm is Relation-like NAT -defined the carrier of (Polynom-Ring R) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cn
m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
Sum mm is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
m1 . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
len m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cn -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
CN - m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
- m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
CN + (- m1) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
(CN - m1) . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
CN . ((len CN) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(CN . ((len CN) -' 1)) - (CN . ((len CN) -' 1)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
ev is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
tcPR . lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
len ev is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(len ev) -' 1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
ev . ((len ev) -' 1) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(len CN) - 1 is V42() V43() V44() ext-real set
((len CN) -' 1) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
((len CN) - 1) + 1 is V42() V43() V44() ext-real set
len (CN - m1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
(CN - m1) + m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
m1 + (- m1) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
CN + (m1 + (- m1)) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
m1 - m1 is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
m1 + (- m1) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
CN + (m1 - m1) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
CN + (0_. R) is Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:]
ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
ev + lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
P is non empty Element of bool the carrier of (Polynom-Ring R)
E is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . E is Element of bool the carrier of (Polynom-Ring R)
LC is non empty Element of bool the carrier of (Polynom-Ring R)
P -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (P -Ideal) is Element of bool the carrier of (Polynom-Ring R)
E + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (E + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . E) \/ {(tcPR . (C + 1))} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,LC) is non empty Element of bool LC
bool LC is non empty set
{ b₁ where b₁ is Element of LC : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in LC or len b₂ <= len b₃ ) } is set
len (- m1) is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
cR -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
(tcPR . (C + 1)) - ev is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring R)
P is non empty Element of bool the carrier of (Polynom-Ring R)
E is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . E is Element of bool the carrier of (Polynom-Ring R)
LC is non empty Element of bool the carrier of (Polynom-Ring R)
P -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (P -Ideal) is Element of bool the carrier of (Polynom-Ring R)
E + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (E + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . E) \/ {(tcPR . (C + 1))} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,LC) is non empty Element of bool LC
bool LC is non empty set
{ b₁ where b₁ is Element of LC : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in LC or len b₂ <= len b₃ ) } is set
cR \ (cn -Ideal) is Element of bool the carrier of (Polynom-Ring R)
ev is non empty Element of bool the carrier of (Polynom-Ring R)
lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
S . lc is Element of bool the carrier of (Polynom-Ring R)
E is non empty Element of bool the carrier of (Polynom-Ring R)
ev -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring R)
cR \ (ev -Ideal) is Element of bool the carrier of (Polynom-Ring R)
lc + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
S . (lc + 1) is Element of bool the carrier of (Polynom-Ring R)
(S . lc) \/ {(tcPR . (C + 1))} is non empty Element of bool the carrier of (Polynom-Ring R)
(R,E) is non empty Element of bool E
bool E is non empty set
{ b₁ where b₁ is Element of E : for b₂, b₃ being Relation-like NAT -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of R:] holds
( not b₂ = b₁ or not b₃ in E or len b₂ <= len b₃ ) } is set
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
Polynom-Ring (Polynom-Ring (X,R)) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (Polynom-Ring (X,R))) is non empty set
X + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
Polynom-Ring ((X + 1),R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring ((X + 1),R)) is non empty set
[: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):] is non empty set
f0 is Relation-like the carrier of (Polynom-Ring (Polynom-Ring (X,R))) -defined the carrier of (Polynom-Ring ((X + 1),R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of (Polynom-Ring (Polynom-Ring (X,R))), the carrier of (Polynom-Ring ((X + 1),R)):]
the carrier of R is non empty non trivial set
Polynom-Ring (0,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of (Polynom-Ring (0,R)) is non empty set
[: the carrier of R, the carrier of (Polynom-Ring (0,R)):] is Relation-like non empty set
bool [: the carrier of R, the carrier of (Polynom-Ring (0,R)):] is non empty set
X is Relation-like the carrier of R -defined the carrier of (Polynom-Ring (0,R)) -valued Function-like non empty total quasi_total Element of bool [: the carrier of R, the carrier of (Polynom-Ring (0,R)):]
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty non trivial set
bool the carrier of R is non empty set
X is non empty add-closed left-ideal right-ideal Element of bool the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element add-closed left-ideal right-ideal Element of bool the carrier of R
f0 is non empty finite Element of bool the carrier of R
f0 -Ideal is non empty add-closed left-ideal right-ideal finitely_generated Element of bool the carrier of R
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(1_ R)} is non empty trivial finite 1 -element Element of bool the carrier of R
f0 is non empty finite Element of bool the carrier of R
f0 -Ideal is non empty add-closed left-ideal right-ideal finitely_generated Element of bool the carrier of R
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element add-closed left-ideal right-ideal Element of bool the carrier of R
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
X is non empty non trivial ordinal non finite set
Polynom-Ring (X,R) is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital V275() V276() V277() V278() doubleLoopStr
the carrier of R is non empty non trivial set
[:X, the carrier of R:] is Relation-like non empty non trivial non finite set
bool [:X, the carrier of R:] is non empty non trivial non finite set
0. R is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
X --> (0. R) is Relation-like X -defined the carrier of R -valued Function-like constant non empty total quasi_total T-Sequence-like Element of bool [:X, the carrier of R:]
{(0. R)} is non empty trivial finite 1 -element set
[:X,{(0. R)}:] is Relation-like non empty non trivial non finite set
Bags X is functional non empty Element of bool (Bags X)
Bags X is non empty set
bool (Bags X) is non empty set
[:(Bags X), the carrier of R:] is Relation-like non empty set
bool [:(Bags X), the carrier of R:] is non empty set
cR is Element of X
(X,cR,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,cR) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (cR,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0_ (X,R)) +* ((X,cR),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
S is Element of X
(X,S,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,S) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (S,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,S),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
bool the carrier of R is non empty set
{ (X,b₁,R) where b₁ is Element of X : b₁ in NAT } is set
the carrier of (Polynom-Ring (X,R)) is non empty set
S is set
C is Element of X
(X,C,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,C) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (C,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0_ (X,R)) +* ((X,C),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
0X is Element of X
(X,0X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
0_ (X,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,0X) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
EmptyBag X is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (0X,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
1_ R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
K409(R) is V102(R) left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0_ (X,R)) +* ((X,0X),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
bool the carrier of (Polynom-Ring (X,R)) is non empty set
S is non empty Element of bool the carrier of (Polynom-Ring (X,R))
S -Ideal is non empty add-closed left-ideal right-ideal Element of bool the carrier of (Polynom-Ring (X,R))
C is non empty finite Element of bool the carrier of (Polynom-Ring (X,R))
C -Ideal is non empty add-closed left-ideal right-ideal finitely_generated Element of bool the carrier of (Polynom-Ring (X,R))
{ H₁(b₁) where b₁ is Element of X : H₂(b₁) in C } is set
c is set
cn is Element of X
(X,cn,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,cn) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (cn,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,cn),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
cn is Element of X
(X,cn,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,cn) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (cn,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,cn),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
the Element of C is Element of C
cn is Element of X
(X,cn,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,cn) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (cn,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,cn),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
cn is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
CN is non empty finite V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of bool NAT
max CN is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() set
mm is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
mm + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
m1 is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
m2 is Element of X
(X,m2,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,m2) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (m2,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,m2),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
lc is Relation-like NAT -defined the carrier of (Polynom-Ring (X,R)) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of C
Sum lc is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
f0 is Relation-like X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:X, the carrier of R:]
f0 +* (m2,(1_ R)) is Relation-like X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:X, the carrier of R:]
[: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):] is non empty set
E is Relation-like NAT -defined [: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of [: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):]
ev is Relation-like X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:X, the carrier of R:]
Polynom-Evaluation (X,R,ev) is Relation-like the carrier of (Polynom-Ring (X,R)) -defined the carrier of R -valued Function-like non empty total quasi_total unity-preserving multiplicative additive RingHomomorphism Element of bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:]
[: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is Relation-like non empty set
bool [: the carrier of (Polynom-Ring (X,R)), the carrier of R:] is non empty set
len lc is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of R
len LC is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
dom LC is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
LC is set
cR is non empty Element of bool the carrier of R
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
E /. i is Element of [: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):]
(E /. i) `2_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(E /. i) `1_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(E /. i) `3_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
y is Element of cR
LC /. LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
LC . i is set
u is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
a is Element of cR
u * a is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(u * a) * v is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(max CN) + 1 is non empty ordinal natural finite cardinal V42() V43() V44() ext-real positive non negative V49() V72() V73() V74() V75() V76() V77() V290() V291() V292() V293() V294() Element of NAT
i is ordinal natural finite cardinal V42() V43() V44() ext-real non negative V49() V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of NAT
LC is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like finite-support LinearCombination of cR
dom LC is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
dom lc is finite V72() V73() V74() V75() V76() V77() V292() V293() V294() Element of bool NAT
E /. i is Element of [: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):]
(E /. i) `2_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
y is Element of C
n is Element of X
(X,n,R) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total finite-Support Element of bool [:(Bags X), the carrier of R:]
(X,n) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support Element of Bags X
(EmptyBag X) +* (n,1) is Relation-like X -defined RAT -valued Function-like total V62() V63() V64() V65() finite-support set
(0_ (X,R)) +* ((X,n),(1_ R)) is Relation-like Bags X -defined the carrier of R -valued Function-like non empty total quasi_total Element of bool [:(Bags X), the carrier of R:]
ev . n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X --> (0. R)) . n is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Support (X,n,R) is functional finite Element of bool (Bags X)
bool (Bags X) is non empty set
{(X,n)} is functional non empty trivial finite 1 -element Element of bool (Bags X)
(Polynom-Evaluation (X,R,ev)) . (X,n,R) is set
eval ((X,n,R),ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,n,R) . (X,n) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
eval ((X,n),ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((X,n,R) . (X,n)) * (eval ((X,n),ev)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(1_ R) * (eval ((X,n),ev)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(1_ R) * (ev . n) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
LC . i is set
(E /. i) `1_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3)) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(E /. i) `3_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(((Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3)) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3))) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
dom (X --> (0. R)) is non empty Element of bool X
bool X is non empty non trivial non finite set
ev . m2 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Support (X,m2,R) is functional finite Element of bool (Bags X)
{(X,m2)} is functional non empty trivial finite 1 -element Element of bool (Bags X)
(Polynom-Evaluation (X,R,ev)) . (X,m2,R) is set
eval ((X,m2,R),ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(X,m2,R) . (X,m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
eval ((X,m2),ev) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((X,m2,R) . (X,m2)) * (eval ((X,m2),ev)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(1_ R) * (eval ((X,m2),ev)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(1_ R) * (ev . m2) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
i is set
LC . i is set
E /. i is Element of [: the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)), the carrier of (Polynom-Ring (X,R)):]
(E /. i) `1_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(E /. i) `2_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
((Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3)) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(E /. i) `3_3 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of (Polynom-Ring (X,R))
(Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(((Polynom-Evaluation (X,R,ev)) . ((E /. i) `1_3)) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `2_3))) * ((Polynom-Evaluation (X,R,ev)) . ((E /. i) `3_3)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(Polynom-Evaluation (X,R,ev)) . (Sum lc) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
Sum LC is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R